Numerical study of a Whitham equation exhibiting both breaking waves and continuous solutions
| dc.contributor.author | Mortell, Michael P. | |
| dc.contributor.author | Mulchrone, Kieran F. | |
| dc.date.accessioned | 2021-04-08T09:21:00Z | |
| dc.date.available | 2021-04-08T09:21:00Z | |
| dc.date.issued | 2021-04-01 | |
| dc.date.updated | 2021-04-08T09:01:19Z | |
| dc.description.abstract | We consider a Whitham equation as an alternative for the Korteweg–de Vries (KdV) equation in which the third derivative is replaced by the integral of a kernel, i.e., ηxxx in the KdV equation is replaced by ∫∞−∞Kν(x−ξ)ηξ(ξ,t)dξ. The kernel Kν(x) satisfies the conditions limν→∞Kν(x) = δ″(x), where δ(x) is the Dirac delta function and limν→0Kν(x) = 0. The questions studied here, by means of numerical examples, are whether adjustment of the parameter ν produces both continuous solutions and shocks of the kernel equation and how well they represent KdV solutions and solutions of the underlying hyperbolic system. A typical example is for resonant forced oscillations in a closed shallow water tank governed by the kernel equation, which are compared with those governed by a partial differential equation. The continuous solutions of the kernel equation associated with frequency dispersion in the KdV equations limit to the shocks of the shallow water equations as ν → 0. Two experimental problems are solved in a single equation. As another example, suitable adjustment of ν in the kernel equation produces solutions reminiscent of a hydraulic and undular bore. | en |
| dc.description.status | Peer reviewed | en |
| dc.description.version | Published Version | en |
| dc.format.mimetype | application/pdf | en |
| dc.identifier.articleid | 45002 | en |
| dc.identifier.citation | Mortell, M. P. and Mulchrone, K. F. (2021) 'Numerical study of a Whitham equation exhibiting both breaking waves and continuous solutions', AIP Advances, 11 (4), 045002 (13 pp). doi: 10.1063/5.0047582 | en |
| dc.identifier.doi | 10.1063/5.0047582 | en |
| dc.identifier.endpage | 13 | en |
| dc.identifier.issn | 2158-3226 | |
| dc.identifier.issued | 4 | en |
| dc.identifier.journaltitle | AIP Advances | en |
| dc.identifier.startpage | 1 | en |
| dc.identifier.uri | https://hdl.handle.net/10468/11182 | |
| dc.identifier.volume | 11 | en |
| dc.language.iso | en | en |
| dc.publisher | American Institute of Physics | en |
| dc.relation.uri | https://aip.scitation.org/doi/abs/10.1063/5.0047582 | |
| dc.rights | © 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). | en |
| dc.rights.uri | http://creativecommons.org/licenses/by/4.0/ | en |
| dc.subject | Integral transforms | en |
| dc.subject | Soliton solutions | en |
| dc.subject | Surface waves | en |
| dc.subject | Fourier analysis | en |
| dc.subject | Oscillating flow | en |
| dc.subject | Korteweg-de Vries equation | en |
| dc.subject | Generalized functions | en |
| dc.subject | Navier Stokes equations | en |
| dc.title | Numerical study of a Whitham equation exhibiting both breaking waves and continuous solutions | en |
| dc.type | Article (peer-reviewed) | en |
